The role of gentle algebras in higher homological algebra

References

[1] C. Amiot, O. Iyama and I. Reiten, Stable categories of Cohen–Macaulay modules and cluster categories, Amer. J. Math. 137 (2015), no. 3, 813–857. 10.1353/ajm.2015.0019Suche in Google Scholar

[2] K. K. Arnesen, R. Laking and D. Pauksztello, Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra, J. Algebra 467 (2016), 1–46. 10.1016/j.jalgebra.2016.07.019Suche in Google Scholar

[3] I. Assem and D. Happel, Generalized tilted algebras of type A n , Comm. Algebra 9 (1981), no. 20, 2101–2125. 10.1080/00927878108822697Suche in Google Scholar

[4] I. Assem and A. Skowroński, Iterated tilted algebras of type 𝐀 ~ n , Math. Z. 195 (1987), no. 2, 269–290. 10.1007/BF01166463Suche in Google Scholar

[5] K. Baur and R. Coelho Simões, A geometric model for the module category of a gentle algebra, Int. Math. Res. Not. IMRN 2021 (2021), no. 15, 11357–11392. 10.1093/imrn/rnz150Suche in Google Scholar

[6] V. Bekkert and H. A. Merklen, Indecomposables in derived categories of gentle algebras, Algebr. Represent. Theory 6 (2003), no. 3, 285–302. 10.1081/AGB-120021885Suche in Google Scholar

[7] T. Brüstle, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım, On the combinatorics of gentle algebras, Canad. J. Math. 72 (2020), no. 6, 1551–1580. 10.4153/S0008414X19000397Suche in Google Scholar

[8] M. C. R. Butler and C. M. Ringel, Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1–2, 145–179. 10.1080/00927878708823416Suche in Google Scholar

[9] I. Çanakçı, D. Pauksztello and S. Schroll, On extensions for gentle algebras, Canad. J. Math. 73 (2021), no. 1, 249–292. 10.4153/S0008414X2000005XSuche in Google Scholar

[10] E. Darpö and O. Iyama, d-representation-finite self-injective algebras, Adv. Math. 362 (2020), Article ID 106932. 10.1016/j.aim.2019.106932Suche in Google Scholar

[11] E. Darpö and T. Kringeland, Classification of the d-representation-finite trivial extensions of quiver algebras, preprint (2021), https://arxiv.org/abs/2103.15380. Suche in Google Scholar

[12] T. Dyckerhoff, G. Jasso and Y. Lekili, The symplectic geometry of higher Auslander algebras: Symmetric products of disks, Forum Math. Sigma 9 (2021), Paper No. e10. 10.1017/fms.2021.2Suche in Google Scholar

[13] R. Ebrahimi and A. Nasr-Isfahani, Higher Auslander’s formula, Int. Math. Res. Not. IMRN 2021 (2021), 10.1093/imrn/rnab219. 10.1093/imrn/rnab219Suche in Google Scholar

[14] D. E. Evans and M. Pugh, The Nakayama automorphism of the almost Calabi–Yau algebras associated to S ⁢ U ⁢ ( 3 ) modular invariants, Comm. Math. Phys. 312 (2012), no. 1, 179–222. 10.1007/s00220-011-1389-4Suche in Google Scholar

[15] C. Geiss, B. Keller and S. Oppermann, n-angulated categories, J. Reine Angew. Math. 675 (2013), 101–120. 10.1515/CRELLE.2011.177Suche in Google Scholar

[16] F. Haiden, L. Katzarkov and M. Kontsevich, Flat surfaces and stability structures, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 247–318. 10.1007/s10240-017-0095-ySuche in Google Scholar

[17] D. Happel, Auslander–Reiten triangles in derived categories of finite-dimensional algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 641–648. 10.1090/S0002-9939-1991-1045137-6Suche in Google Scholar

[18] J. Haugland, The Grothendieck group of an n-exangulated category, Appl. Categ. Structures 29 (2021), no. 3, 431–446. 10.1007/s10485-020-09622-wSuche in Google Scholar

[19] J. Haugland and M. H. Sandøy, Higher Koszul duality and connections with n-hereditary algebras, preprint (2021), https://arxiv.org/abs/2101.12743. Suche in Google Scholar

[20] M. Herschend and O. Iyama, n-representation-finite algebras and twisted fractionally Calabi–Yau algebras, Bull. Lond. Math. Soc. 43 (2011), no. 3, 449–466. 10.1112/blms/bdq101Suche in Google Scholar

[21] M. Herschend and O. Iyama, Selfinjective quivers with potential and 2-representation-finite algebras, Compos. Math. 147 (2011), no. 6, 1885–1920. 10.1112/S0010437X11005367Suche in Google Scholar

[22] M. Herschend, O. Iyama, H. Minamoto and S. Oppermann, Representation theory of Geigle–Lenzing complete intersections, preprint (2014), https://arxiv.org/abs/1409.0668; to appear in Mem. Amer. Math. Soc. Suche in Google Scholar

[23] M. Herschend, O. Iyama and S. Oppermann, n-representation infinite algebras, Adv. Math. 252 (2014), 292–342. 10.1016/j.aim.2013.09.023Suche in Google Scholar

[24] M. Herschend, P. Jørgensen and L. Vaso, Wide subcategories of d-cluster tilting subcategories, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2281–2309. 10.1090/tran/8051Suche in Google Scholar

[25] M. Herschend, Y. Liu and H. Nakaoka, n-exangulated categories (I): Definitions and fundamental properties, J. Algebra 570 (2021), 531–586. 10.1016/j.jalgebra.2020.11.017Suche in Google Scholar

[26] O. Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51–82. 10.1016/j.aim.2006.06.003Suche in Google Scholar

[27] O. Iyama, Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50. 10.1016/j.aim.2006.06.002Suche in Google Scholar

[28] O. Iyama, Auslander–Reiten theory revisited, Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2008), 349–397. 10.4171/062-1/8Suche in Google Scholar

[29] O. Iyama, Cluster tilting for higher Auslander algebras, Adv. Math. 226 (2011), no. 1, 1–61. 10.1016/j.aim.2010.03.004Suche in Google Scholar

[30] O. Iyama and S. Oppermann, n-representation-finite algebras and n-APR tilting, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6575–6614. 10.1090/S0002-9947-2011-05312-2Suche in Google Scholar

[31] O. Iyama and S. Oppermann, Stable categories of higher preprojective algebras, Adv. Math. 244 (2013), 23–68. 10.1016/j.aim.2013.03.013Suche in Google Scholar

[32] O. Iyama and M. Wemyss, Maximal modifications and Auslander–Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521–586. 10.1007/s00222-013-0491-ySuche in Google Scholar

[33] O. Iyama and Y. Yoshino, Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168. 10.1007/s00222-007-0096-4Suche in Google Scholar

[34] K. M. Jacobsen and P. Jørgensen, d-abelian quotients of ( d + 2 ) -angulated categories, J. Algebra 521 (2019), 114–136. 10.1016/j.jalgebra.2018.11.019Suche in Google Scholar

[35] K. M. Jacobsen and P. Jørgensen, Maximal τ d -rigid pairs, J. Algebra 546 (2020), 119–134. 10.1016/j.jalgebra.2019.10.046Suche in Google Scholar

[36] G. Jasso, n-abelian and n-exact categories, Math. Z. 283 (2016), no. 3–4, 703–759. 10.1007/s00209-016-1619-8Suche in Google Scholar

[37] G. Jasso and J. Külshammer, Higher Nakayama algebras I: Construction, Adv. Math. 351 (2019), 1139–1200. 10.1016/j.aim.2019.05.026Suche in Google Scholar

[38] G. Jasso and S. Kvamme, An introduction to higher Auslander–Reiten theory, Bull. Lond. Math. Soc. 51 (2019), no. 1, 1–24. 10.1112/blms.12204Suche in Google Scholar

[39] P. Jørgensen, Torsion classes and t-structures in higher homological algebra, Int. Math. Res. Not. IMRN 2016 (2016), no. 13, 3880–3905. 10.1093/imrn/rnv265Suche in Google Scholar

[40] P. Jørgensen, Tropical friezes and the index in higher homological algebra, Math. Proc. Cambridge Philos. Soc. 171 (2021), no. 1, 23–49. 10.1017/S0305004120000031Suche in Google Scholar

[41] B. Keller and I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math. 211 (2007), no. 1, 123–151. 10.1016/j.aim.2006.07.013Suche in Google Scholar

[42] S. Kvamme, Axiomatizing subcategories of Abelian categories, J. Pure Appl. Algebra 226 (2022), no. 4, Paper No. 106862. 10.1016/j.jpaa.2021.106862Suche in Google Scholar

[43] Y. Lekili and A. Polishchuk, Derived equivalences of gentle algebras via Fukaya categories, Math. Ann. 376 (2020), no. 1–2, 187–225. 10.1007/s00208-019-01894-5Suche in Google Scholar

[44] Y. Mizuno, A Gabriel-type theorem for cluster tilting, Proc. Lond. Math. Soc. (3) 108 (2014), no. 4, 836–868. 10.1112/plms/pdt046Suche in Google Scholar

[45] S. Opper, P.-G. Plamondon and S. Schroll, A geometric model for the derived category of gentle algebras, preprint (2018), https://arxiv.org/abs/1801.09659. Suche in Google Scholar

[46] S. Oppermann, C. Psaroudakis and T. Stai, Partial serre duality and cocompact objects, preprint (2021), https://arxiv.org/abs/2104.12498. Suche in Google Scholar

[47] S. Oppermann and H. Thomas, Higher-dimensional cluster combinatorics and representation theory, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1679–1737. 10.4171/JEMS/345Suche in Google Scholar

[48] Y. Palu, V. Pilaud and P.-G. Plamondon, Non-kissing and non-crossing complexes for locally gentle algebras, J. Comb. Algebra 3 (2019), no. 4, 401–438. 10.4171/JCA/35Suche in Google Scholar

[49] J. Reid, Indecomposable objects determined by their index in higher homological algebra, Proc. Amer. Math. Soc. 148 (2020), no. 6, 2331–2343. 10.1090/proc/14924Suche in Google Scholar

[50] M. H. Sandøy and L.-P. Thibault, Classification results for n-hereditary monomial algebras, preprint (2021), https://arxiv.org/abs/2101.12746. Suche in Google Scholar

[51] J. Schröer, Modules without self-extensions over gentle algebras, J. Algebra 216 (1999), no. 1, 178–189. 10.1006/jabr.1998.7696Suche in Google Scholar

[52] S. Schroll, Trivial extensions of gentle algebras and Brauer graph algebras, J. Algebra 444 (2015), 183–200. 10.1016/j.jalgebra.2015.07.037Suche in Google Scholar

[53] S. Schroll, Brauer graph algebras: a survey on Brauer graph algebras, associated gentle algebras and their connections to cluster theory, Homological Methods, Representation Theory, and Cluster Algebras, CRM Short Courses, Springer, Cham (2018), 177–223. 10.1007/978-3-319-74585-5_6Suche in Google Scholar

[54] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts 65, Cambridge University, Cambridge, 2006. 10.1017/CBO9780511614309Suche in Google Scholar

[55] L. Vaso, n-cluster tilting subcategories of representation-directed algebras, J. Pure Appl. Algebra 223 (2019), no. 5, 2101–2122. 10.1016/j.jpaa.2018.07.010Suche in Google Scholar

[56] L. Vaso, n-cluster tilting subcategories for radical square zero algebras, preprint (2021), https://arxiv.org/abs/2105.05830. 10.1016/j.jpaa.2022.107157Suche in Google Scholar

[57] N. J. Williams, New interpretations of the higher Stasheff–Tamari orders, preprint (2020), https://arxiv.org/abs/2007.12664. 10.1016/j.aim.2022.108552Suche in Google Scholar